YES
0 QTRS
↳1 Overlay + Local Confluence (⇔, 24 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 0 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔, 0 ms)
↳9 QDP
↳10 ATransformationProof (⇔, 0 ms)
↳11 QDP
↳12 QReductionProof (⇔, 0 ms)
↳13 QDP
↳14 QDPSizeChangeProof (⇔, 0 ms)
↳15 YES
↳16 QDP
↳17 UsableRulesProof (⇔, 0 ms)
↳18 QDP
↳19 ATransformationProof (⇔, 0 ms)
↳20 QDP
↳21 QReductionProof (⇔, 0 ms)
↳22 QDP
↳23 QDPSizeChangeProof (⇔, 0 ms)
↳24 YES
↳25 QDP
↳26 UsableRulesProof (⇔, 0 ms)
↳27 QDP
↳28 QDPSizeChangeProof (⇔, 0 ms)
↳29 YES
app(f, app(s, x)) → app(f, x)
app(g, app(app(cons, 0), y)) → app(g, y)
app(g, app(app(cons, app(s, x)), y)) → app(s, x)
app(h, app(app(cons, x), y)) → app(h, app(g, app(app(cons, x), y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(f, app(s, x)) → app(f, x)
app(g, app(app(cons, 0), y)) → app(g, y)
app(g, app(app(cons, app(s, x)), y)) → app(s, x)
app(h, app(app(cons, x), y)) → app(h, app(g, app(app(cons, x), y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(f, app(s, x0))
app(g, app(app(cons, 0), x0))
app(g, app(app(cons, app(s, x0)), x1))
app(h, app(app(cons, x0), x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
APP(f, app(s, x)) → APP(f, x)
APP(g, app(app(cons, 0), y)) → APP(g, y)
APP(h, app(app(cons, x), y)) → APP(h, app(g, app(app(cons, x), y)))
APP(h, app(app(cons, x), y)) → APP(g, app(app(cons, x), y))
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(cons, app(fun, x)), app(app(map, fun), xs))
APP(app(map, fun), app(app(cons, x), xs)) → APP(cons, app(fun, x))
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(filter2, app(fun, x)), fun), x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(filter2, app(fun, x)), fun)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(filter2, app(fun, x))
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(cons, x), app(app(filter, fun), xs))
APP(app(app(app(filter2, true), fun), x), xs) → APP(cons, x)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, true), fun), x), xs) → APP(filter, fun)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, false), fun), x), xs) → APP(filter, fun)
app(f, app(s, x)) → app(f, x)
app(g, app(app(cons, 0), y)) → app(g, y)
app(g, app(app(cons, app(s, x)), y)) → app(s, x)
app(h, app(app(cons, x), y)) → app(h, app(g, app(app(cons, x), y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(f, app(s, x0))
app(g, app(app(cons, 0), x0))
app(g, app(app(cons, app(s, x0)), x1))
app(h, app(app(cons, x0), x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
APP(g, app(app(cons, 0), y)) → APP(g, y)
app(f, app(s, x)) → app(f, x)
app(g, app(app(cons, 0), y)) → app(g, y)
app(g, app(app(cons, app(s, x)), y)) → app(s, x)
app(h, app(app(cons, x), y)) → app(h, app(g, app(app(cons, x), y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(f, app(s, x0))
app(g, app(app(cons, 0), x0))
app(g, app(app(cons, app(s, x0)), x1))
app(h, app(app(cons, x0), x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
APP(g, app(app(cons, 0), y)) → APP(g, y)
app(f, app(s, x0))
app(g, app(app(cons, 0), x0))
app(g, app(app(cons, app(s, x0)), x1))
app(h, app(app(cons, x0), x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
g1(cons(0, y)) → g1(y)
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
g1(cons(0, y)) → g1(y)
From the DPs we obtained the following set of size-change graphs:
APP(f, app(s, x)) → APP(f, x)
app(f, app(s, x)) → app(f, x)
app(g, app(app(cons, 0), y)) → app(g, y)
app(g, app(app(cons, app(s, x)), y)) → app(s, x)
app(h, app(app(cons, x), y)) → app(h, app(g, app(app(cons, x), y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(f, app(s, x0))
app(g, app(app(cons, 0), x0))
app(g, app(app(cons, app(s, x0)), x1))
app(h, app(app(cons, x0), x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
APP(f, app(s, x)) → APP(f, x)
app(f, app(s, x0))
app(g, app(app(cons, 0), x0))
app(g, app(app(cons, app(s, x0)), x1))
app(h, app(app(cons, x0), x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
f1(s(x)) → f1(x)
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
f(s(x0))
g(cons(0, x0))
g(cons(s(x0), x1))
h(cons(x0, x1))
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
f1(s(x)) → f1(x)
From the DPs we obtained the following set of size-change graphs:
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
app(f, app(s, x)) → app(f, x)
app(g, app(app(cons, 0), y)) → app(g, y)
app(g, app(app(cons, app(s, x)), y)) → app(s, x)
app(h, app(app(cons, x), y)) → app(h, app(g, app(app(cons, x), y)))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(f, app(s, x0))
app(g, app(app(cons, 0), x0))
app(g, app(app(cons, app(s, x0)), x1))
app(h, app(app(cons, x0), x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
app(f, app(s, x0))
app(g, app(app(cons, 0), x0))
app(g, app(app(cons, app(s, x0)), x1))
app(h, app(app(cons, x0), x1))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
From the DPs we obtained the following set of size-change graphs: